MATH 131-505 Spring 2015
2.7
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2.7 The Derivative as a Function
The derivative of f is defined as
f (x + h) − f (x)
h→0
h
f 0 (x) = lim
if the limit exists.
Examples:
1. (p. 146) The graph of a function f is given below. Use it to sketch the graph of f 0 . Find f (1)
and f 0 (1).
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MATH 131-505 Spring 2015
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c
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Liu
√
2. (p. 156) f (x) = 1 + 2x
(a) find the derivative of f using the definition of derivative.
(b) State the domain of f and f 0 .
Other notations: If we use the traditional notation y = f (x) to indicate that the independent
variable is x and the dependent variable is y, then some common alternative notations for the
derivative are as follows:
dy
df
d
f 0 (x) = y 0 =
=
=
f (x) = Df (x) = Dx f (x)
dx
dx
dx
d
are called differentiation operators because they indicate the operation
The symbols D and
dx
of differentiation, which is the process of calculating a derivative.
dy
If we want to indicate the value of a derivative
at a specific number a, we use the notation
dx
dy dy
0
0
f (a) =
or f (a) =
dx x=a
dx x=a
A function f is differentiable at a if f 0 (a) exists. It is differentiable on an open interval (a, b)
(or (a, ∞) or (−∞, a) or (−∞, ∞)) if it is differentiable at every number in the interval.
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MATH 131-505 Spring 2015
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Example 3: (p. 150) Where is the function f (x) = |x| differentiable?
Theorem: If f is differentiable at a, then f is continuous at a.
Note: The converse of the theorem is false; that is, there are functions that are continuous but not
differentiable (see example 3).
How Can a Function Fail to be Differentiable? (p. 152)
The last two graphs show that the curve has a vertical tangent line when x = a; that is, f is
continuous at a and lim |f 0 (x)| = ∞.
x→a
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MATH 131-505 Spring 2015
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2.7
Example 4: (p. 155) Match the graph of each function in the second row with the graph of its
derivative in the first row. Give reasons for your choices.
(a)
(b)
(c)
(d)
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MATH 131-505 Spring 2015
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Liu
2.7
Higher Derivatives
If f is a differentiable function, then its derivative f 0 is also a function, so f 0 may have a derivative of
its own, denoted by (f 0 )0 = f 00 . This new function f 00 is called the second derivative of f because
it is the derivative of the derivative of f . Using Leibniz notation,
d dy
d2 y
= 2
dx dx
dx
which can be interpreted as the slope of the curve y = f 0 (x) at the point (x, f 0 (x)). In other words,
it is the rate of change of the slope of the original curve y = f (x).
In general, the nth derivative of f is denoted by f (n) and is obtained from f by differentiating n
times. If y = f (x), we write
dn y
y (n) = f (n) (x) = n
dx
Applications: If s = s(t) is the position function of an object that moves in a straight line, its first
derivative represents the velocity v(t) of the object
v(t) = s0 (t) =
ds
dt
The instantaneous rate of change of velocity with respect to time is called the acceleration a(t) of
the object
d2 s
a(t) = v 0 (t) = s00 (t) = 2
dt
Examples:
5. (p. 154) If f (x) = x3 − x, find f 00 (2), f 000 (1), and f (4) (a) for some constant a.
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MATH 131-505 Spring 2015
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6. (p. 157) The figure shows the graphs of four functions. One is the position function of a car, one
is the velocity of the car, one is its acceleration, and one is its jerk. Identify each curve and explain
your choices.
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